#72259918

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#455 ⋊ C₂×C₃
Hall	#11737737 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #3019
Sylow	C₇ ⋊ C₃×#3019

Subgroups

	subgroup	G/H
center	C₂	#120×AGL(1,𝔽₇)
commutator, cosocle	C₂×C₄×C₇	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₇)
Fitting	C₇×#455	C₂×C₃
socle	C₂×C₇	C₂×C₃×#120
hypercenter	#455	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#3019
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #455

1524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 112

Polycyclic presentation:

2^4,3,2^3,7|c^2=h,f^2=h,b^a=bh,c^b=cf,d^b=dg,d^c=dh,f^b=fh,f^c=fh,f^d=fh,g^c=gh,i^a=i^6,i^b=i^6,i^e=i^2

Conjugacy classes & character table

103 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72259922

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 2, 2, 2, 7]);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^6);;
gap> SetConjugate(coll, 9, 2, F.9^6);;
gap> SetConjugate(coll, 9, 5, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);